Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

Abstract

In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”

“…Ilyin [2] considers the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared with the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly.…”

“…Numerical simulation in physical, social, and life sciences [1][2][3][4]; • Modeling and analysis of complex systems based on mathematical methods and AI/ML approaches [5,6]; • Control problems in robotics [3,[7][8][9][10][11][12]]; • Design optimization of complex systems [13]; • Modeling in economics and social sciences [4,14]; • Stochastic models in physics and engineering [1,[15][16][17][18]; • Mathematical models in material science [19]; • High-performance computing for mathematical modeling [20].…”